We proved the (local) path-connectedness of certain subset of the singular set of semiconcave functions with linear modulus in general. In some sense this result is optimal. The proof is based on a theorem by Marie-Claude Arnaud (M.-C. Arnaud, textit{Pseudographs and the Lax-Oleinik semi-group: a geometric and dynamical interpretation}. Nonlinearity, textbf{24}(1): 71-78, 2011.). We also gave a new proof of the theorem in time-dependent case.
{"title":"Topology of singular set of semiconcave function via Arnaud's theorem","authors":"Tianqi Shi, Wei Cheng, Jiahui Hong","doi":"10.3934/cpaa.2023053","DOIUrl":"https://doi.org/10.3934/cpaa.2023053","url":null,"abstract":"We proved the (local) path-connectedness of certain subset of the singular set of semiconcave functions with linear modulus in general. In some sense this result is optimal. The proof is based on a theorem by Marie-Claude Arnaud (M.-C. Arnaud, textit{Pseudographs and the Lax-Oleinik semi-group: a geometric and dynamical interpretation}. Nonlinearity, textbf{24}(1): 71-78, 2011.). We also gave a new proof of the theorem in time-dependent case.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41881565","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. In this paper, we consider the homogenization problem for generalized elliptic systems L ε = − div( A ( x/ε ) ∇ + V ( x/ε )) + B ( x/ε ) ∇ + c ( x/ε ) + λI with dimension two. Precisely, we will establish the W 1 ,p estimates, H¨older estimates, Lipschitz estimates and L p convergence results for L ε with dimension two. The operator L ε has been studied by Qiang Xu with dimension d ≥ 3 in [22, 23] and the case d = 2 is remained unsolved. As a byproduct, we will construct the Green functions for L ε with d = 2 and their convergence rates.
{"title":"Homogenization theory of elliptic system with lower order terms for dimension two","authors":"Wen Wang, Ting Zhang","doi":"10.3934/cpaa.2023010","DOIUrl":"https://doi.org/10.3934/cpaa.2023010","url":null,"abstract":". In this paper, we consider the homogenization problem for generalized elliptic systems L ε = − div( A ( x/ε ) ∇ + V ( x/ε )) + B ( x/ε ) ∇ + c ( x/ε ) + λI with dimension two. Precisely, we will establish the W 1 ,p estimates, H¨older estimates, Lipschitz estimates and L p convergence results for L ε with dimension two. The operator L ε has been studied by Qiang Xu with dimension d ≥ 3 in [22, 23] and the case d = 2 is remained unsolved. As a byproduct, we will construct the Green functions for L ε with d = 2 and their convergence rates.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43923183","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We compute low energy asymptotics for the resolvent of a planar obstacle, and deduce asymptotics for the corresponding scattering matrix, scattering phase, and exterior Dirichlet-to-Neumann operator. We use an identity of Vodev to relate the obstacle resolvent to the free resolvent and an identity of Petkov and Zworski to relate the scattering matrix to the resolvent. The leading singularities are given in terms of the obstacle's logarithmic capacity or Robin constant. We expect these results to hold for more general compactly supported perturbations of the Laplacian on $mathbb R^2$, with the definition of the Robin constant suitably modified, under a generic assumption that the spectrum is regular at zero.
{"title":"Low energy scattering asymptotics for planar obstacles","authors":"T. Christiansen, K. Datchev","doi":"10.2140/paa.2023.5.767","DOIUrl":"https://doi.org/10.2140/paa.2023.5.767","url":null,"abstract":"We compute low energy asymptotics for the resolvent of a planar obstacle, and deduce asymptotics for the corresponding scattering matrix, scattering phase, and exterior Dirichlet-to-Neumann operator. We use an identity of Vodev to relate the obstacle resolvent to the free resolvent and an identity of Petkov and Zworski to relate the scattering matrix to the resolvent. The leading singularities are given in terms of the obstacle's logarithmic capacity or Robin constant. We expect these results to hold for more general compactly supported perturbations of the Laplacian on $mathbb R^2$, with the definition of the Robin constant suitably modified, under a generic assumption that the spectrum is regular at zero.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89331783","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we establish the gradient and Pogorelov estimates for $k$-convex-monotone solutions to parabolic $k$-Hessian equations of the form $-u_tsigma_k(lambda(D^2u))=psi(x,t,u)$. We also apply such estimates to obtain a Liouville type result, which states that any $k$-convex-monotone and $C^{4,2}$ solution $u$ to $-u_tsigma_k(lambda(D^2u))=1$ in $mathbb{R}^ntimes(-infty,0]$ must be a linear function of $t$ plus a quadratic polynomial of $x$, under some growth assumptions on $u$.
{"title":"Interior estimates of derivatives and a Liouville type theorem for parabolic $ k $-Hessian equations","authors":"J. Bao, J. Qiang, Z. Tang, C. Wang","doi":"10.3934/cpaa.2023073","DOIUrl":"https://doi.org/10.3934/cpaa.2023073","url":null,"abstract":"In this paper, we establish the gradient and Pogorelov estimates for $k$-convex-monotone solutions to parabolic $k$-Hessian equations of the form $-u_tsigma_k(lambda(D^2u))=psi(x,t,u)$. We also apply such estimates to obtain a Liouville type result, which states that any $k$-convex-monotone and $C^{4,2}$ solution $u$ to $-u_tsigma_k(lambda(D^2u))=1$ in $mathbb{R}^ntimes(-infty,0]$ must be a linear function of $t$ plus a quadratic polynomial of $x$, under some growth assumptions on $u$.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41495155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the Dirichlet problems for second order linear elliptic equations in non-divergence and divergence forms on a bounded domain $Omega$ in $mathbb{R}^n$, $n ge 2$: $$ -sum_{i,j=1}^n a^{ij}D_{ij} u + b cdot D u + cu = f ;;text{ in $Omega$} quad text{and} quad u=0 ;;text{ on $partial Omega$} $$ and $$ - {rm div} left( A D u right) + {rm div}(ub) + cu = {rm div} F ;;text{ in $Omega$} quad text{and} quad u=0 ;;text{ on $partial Omega$} , $$ where $A=[a^{ij}]$ is symmetric, uniformly elliptic, and of vanishing mean oscillation (VMO). The main purposes of this paper is to study unique solvability for both problems with $L^1$-data. We prove that if $Omega$ is of class $C^{1}$, $ {rm div} A + bin L^{n,1}(Omega;mathbb{R}^n)$, $cin L^{frac{n}{2},1}(Omega) cap L^s(Omega)$ for some $1
在$mathbb{R}^n$, $n ge 2$: $$ -sum_{i,j=1}^n a^{ij}D_{ij} u + b cdot D u + cu = f ;;text{ in $Omega$} quad text{and} quad u=0 ;;text{ on $partial Omega$} $$和$$ - {rm div} left( A D u right) + {rm div}(ub) + cu = {rm div} F ;;text{ in $Omega$} quad text{and} quad u=0 ;;text{ on $partial Omega$} , $$的有界区域$Omega$上，考虑二阶线性椭圆方程的非散度和散度形式的Dirichlet问题，其中$A=[a^{ij}]$是对称的，均匀椭圆的，并且具有消失的平均振荡(VMO)。本文的主要目的是研究$L^1$ -数据下这两个问题的唯一可解性。我们证明了如果$Omega$是$C^{1}$, $ {rm div} A + bin L^{n,1}(Omega;mathbb{R}^n)$, $cin L^{frac{n}{2},1}(Omega) cap L^s(Omega)$对于$1
{"title":"Dirichlet problems for second order linear elliptic equations with $ L^{1} $-data","authors":"Hyunseok Kim, Jisu Oh","doi":"10.3934/cpaa.2023051","DOIUrl":"https://doi.org/10.3934/cpaa.2023051","url":null,"abstract":"We consider the Dirichlet problems for second order linear elliptic equations in non-divergence and divergence forms on a bounded domain $Omega$ in $mathbb{R}^n$, $n ge 2$: $$ -sum_{i,j=1}^n a^{ij}D_{ij} u + b cdot D u + cu = f ;;text{ in $Omega$} quad text{and} quad u=0 ;;text{ on $partial Omega$} $$ and $$ - {rm div} left( A D u right) + {rm div}(ub) + cu = {rm div} F ;;text{ in $Omega$} quad text{and} quad u=0 ;;text{ on $partial Omega$} , $$ where $A=[a^{ij}]$ is symmetric, uniformly elliptic, and of vanishing mean oscillation (VMO). The main purposes of this paper is to study unique solvability for both problems with $L^1$-data. We prove that if $Omega$ is of class $C^{1}$, $ {rm div} A + bin L^{n,1}(Omega;mathbb{R}^n)$, $cin L^{frac{n}{2},1}(Omega) cap L^s(Omega)$ for some $1<s<frac{3}{2}$, and $cge0$ in $Omega$, then for each $fin L^1 (Omega )$, there exists a unique weak solution in $W^{1,frac{n}{n-1},infty}_0 (Omega)$ of the first problem. Moreover, under the additional condition that $Omega$ is of class $C^{1,1}$ and $cin L^{n,1}(Omega)$, we show that for each $F in L^1 (Omega ; mathbb{R}^n)$, the second problem has a unique very weak solution in $L^{frac{n}{n-1},infty}(Omega)$.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44789652","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper generalizes and extends to the case of nonlinear effects and logistic perturbations some results recently developed in the literature where, for the linear counterpart and in absence of logistics, criteria toward boundedness for an attraction-repulsion Keller-Segel system with double saturation are derived.
{"title":"A nonlinear attraction-repulsion Keller–Segel model with double sublinear absorptions: criteria toward boundedness","authors":"Yutaro Chiyo, Silvia Frassu, G. Viglialoro","doi":"10.3934/cpaa.2023047","DOIUrl":"https://doi.org/10.3934/cpaa.2023047","url":null,"abstract":"This paper generalizes and extends to the case of nonlinear effects and logistic perturbations some results recently developed in the literature where, for the linear counterpart and in absence of logistics, criteria toward boundedness for an attraction-repulsion Keller-Segel system with double saturation are derived.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48399804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study two topologies τKR and τK on the space of measures on a completely regular space generated by Kantorovich–Rubinshtein and Kantorovich seminorms analogous to their classical norms in the case of a metric space. The Kantorovich–Rubinshtein topology τKR coincides with the weak topology on nonnegative measures and on bounded uniformly tight sets of measures. A sufficient condition is given for the compactness in the Kantorovich topology. We show that for logarithmically concave measures and stable measures weak convergence implies convergence in the Kantorovich topology. We also obtain an efficiently verified condition for convergence of the barycenters of Radon measures from a sequence or net weakly converging on a locally convex space. As an application it is shown that for weakly convergent logarithmically concave measures and stable measures convergence of their barycenters holds without additional conditions. The same is true for measures given by polynomial densities of a fixed degree with respect to logarithmically concave measures.
{"title":"Kantorovich type topologies on spaces of measures and convergence of barycenters","authors":"K. A. Afonin, V. Bogachev","doi":"10.3934/cpaa.2023002","DOIUrl":"https://doi.org/10.3934/cpaa.2023002","url":null,"abstract":"We study two topologies τKR and τK on the space of measures on a completely regular space generated by Kantorovich–Rubinshtein and Kantorovich seminorms analogous to their classical norms in the case of a metric space. The Kantorovich–Rubinshtein topology τKR coincides with the weak topology on nonnegative measures and on bounded uniformly tight sets of measures. A sufficient condition is given for the compactness in the Kantorovich topology. We show that for logarithmically concave measures and stable measures weak convergence implies convergence in the Kantorovich topology. We also obtain an efficiently verified condition for convergence of the barycenters of Radon measures from a sequence or net weakly converging on a locally convex space. As an application it is shown that for weakly convergent logarithmically concave measures and stable measures convergence of their barycenters holds without additional conditions. The same is true for measures given by polynomial densities of a fixed degree with respect to logarithmically concave measures.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43027371","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Juli'an L'opez-G'omez, Eduardo Munoz-Hern'andez, F. Zanolin
This paper studies the global structure of the set of nodal solutions of a generalized Sturm--Liouville boundary value problem associated to the quasilinear equation $$ -(phi(u'))'= lambda u + a(t)g(u), quad lambdain {mathbb R}, $$ where $a(t)$ is non-negative with some positive humps separated away by intervals of degeneracy where $aequiv 0$. When $phi(s)=s$ this equation includes a generalized prototype of a classical model going back to Moore and Nehari, 1959. This is the first paper where the general case when $lambdainmathbb{R}$ has been addressed when $agneq 0$. The semilinear case with $alneq 0$ has been recently treated by L'{o}pez-G'{o}mez and Rabinowitz.
{"title":"Rich dynamics in planar systems with heterogeneous nonnegative weights","authors":"Juli'an L'opez-G'omez, Eduardo Munoz-Hern'andez, F. Zanolin","doi":"10.3934/cpaa.2023020","DOIUrl":"https://doi.org/10.3934/cpaa.2023020","url":null,"abstract":"This paper studies the global structure of the set of nodal solutions of a generalized Sturm--Liouville boundary value problem associated to the quasilinear equation $$ -(phi(u'))'= lambda u + a(t)g(u), quad lambdain {mathbb R}, $$ where $a(t)$ is non-negative with some positive humps separated away by intervals of degeneracy where $aequiv 0$. When $phi(s)=s$ this equation includes a generalized prototype of a classical model going back to Moore and Nehari, 1959. This is the first paper where the general case when $lambdainmathbb{R}$ has been addressed when $agneq 0$. The semilinear case with $alneq 0$ has been recently treated by L'{o}pez-G'{o}mez and Rabinowitz.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41569964","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study semilinear problems in bounded C1,1 domains for non-local operators with a boundary condition. The operators cover and extend the case of the spectral fractional Laplacian. We also study harmonic functions with respect to the nonlocal operator and boundary behaviour of Green and Poisson potentials. AMS 2020 Mathematics Subject Classification: Primary 35J61, 35R11; Secondary 35C15, 31B10, 31B25, 31C05, 60J35
{"title":"Semilinear Dirichlet problem for subordinate spectral Laplacian","authors":"I. Biočić","doi":"10.3934/cpaa.2023012","DOIUrl":"https://doi.org/10.3934/cpaa.2023012","url":null,"abstract":"We study semilinear problems in bounded C1,1 domains for non-local operators with a boundary condition. The operators cover and extend the case of the spectral fractional Laplacian. We also study harmonic functions with respect to the nonlocal operator and boundary behaviour of Green and Poisson potentials. AMS 2020 Mathematics Subject Classification: Primary 35J61, 35R11; Secondary 35C15, 31B10, 31B25, 31C05, 60J35","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46788234","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the stationary problem for a diffusive logistic equation with the homogeneous Dirichlet boundary condition. Concerning the corresponding Neumann problem, Wei-Ming Ni proposed a question as follows: Maximizing the ratio of the total masses of species to resources. For this question, Bai, He and Li showed that the supremum of the ratio is 3 in the one dimensional case, and the author and Kuto showed that the supremum is infinity in the multi-dimensional ball. In this paper, we show the same results still hold true for the Dirichlet problem. Our proof is based on the sub-super solution method and needs more delicate calculation because of the range of the diffusion rate for the existence of the solution.
{"title":"On the ratio of total masses of species to resources for a logistic equation with Dirichlet boundary condition","authors":"Jumpei Inoue","doi":"10.3934/cpaa.2023009","DOIUrl":"https://doi.org/10.3934/cpaa.2023009","url":null,"abstract":"We consider the stationary problem for a diffusive logistic equation with the homogeneous Dirichlet boundary condition. Concerning the corresponding Neumann problem, Wei-Ming Ni proposed a question as follows: Maximizing the ratio of the total masses of species to resources. For this question, Bai, He and Li showed that the supremum of the ratio is 3 in the one dimensional case, and the author and Kuto showed that the supremum is infinity in the multi-dimensional ball. In this paper, we show the same results still hold true for the Dirichlet problem. Our proof is based on the sub-super solution method and needs more delicate calculation because of the range of the diffusion rate for the existence of the solution.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42696911","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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